cert
plain-language theorem explainer
Packages the three cost and threshold facts that underwrite electron spin s = 1/2 from config dimension D = 2 into a single certificate. Anyone citing the structural spin-from-ConfigDim result reaches for this inhabited record. Construction is pure field assignment from three sibling lemmas already proved in-module.
Claim. There is a certificate asserting: (i) the domain cost vanishes on the diagonal, $\mathrm{cost}(r,r)=0$ for all $r\neq 0$; (ii) domain cost is nonnegative for positive arguments; (iii) the canonical threshold is strictly positive. Together these underwrite $s=1/2=1/D$ with $D=2$ for the spinor representation.
background
The module treats electron spin as a structural consequence of config dimension: $s=1/2=1/D$ with $D=2$ for the spinor representation of SU(2) (Clifford algebra in two dimensions). In Recognition Science language, spin emerges from the $D=2$ binary recognition lattice, equivalently $s=1/2^1$.
Domain cost is the local cost functional on mass/energy-like pairs; the certificate demands it vanish when the two arguments agree (nonzero) and stay nonnegative when both are positive. The canonical threshold is the positive cutoff against which that cost is compared. Upstream, nonnegativity of recognition-event cost is already known from ObserverForcing via $J$-cost nonnegativity.
proof idea
One-line structure inhabitant. The three fields of SpinQuantumNumCert are filled by the in-module lemmas domainCost_at_eq, domainCost_nonneg, and canonicalThreshold_pos respectively. No further tactic work; the certificate is just the bundled record of those three facts.
why it matters
Closes the certificate side of the structural (0-sorry, 0-axiom) claim that electron spin $s=1/2$ is forced by config dimension $D=2$, not put in by hand. That claim sits in the physics layer that reads spin off the binary recognition lattice, consistent with the forcing-chain picture in which spatial $D=3$ is forced at T8 while the spinor sector remains the $D=2$ Clifford piece. No downstream consumers are wired yet in the graph; the immediate sibling cert_inhabited is the natural next use. The certificate does not itself derive the mass ladder or $\alpha$ band; it only packages the cost/threshold side conditions the spin reading needs.
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